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What this talk is about? Technically: new understanding of a basic and important family of codes Conceptually: structure and pseudo-randomness play important roles in many computational domains. This talk shows this phenomena applied to coding theory

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Decoding from errors The basic problem of coding theory: recovering from errors Goal: recover correct codeword from a noisy received word This work: worst-case errors

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Unique decoding Codeword Received word

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Unique decoding Unique decoding: find the closest codeword Basic limitation: minimal distance of the code If a received word is “in between” two codewords, we cannot distinguish which is the correct codeword Limits error to

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Polynomial codes Most codes are based on polynomials In this talk, focus on the most basic families Reed-Solomon: univariate polynomials Reed-Muller: multivariate polynomials Despite (or because) being basic, they are widely applied; however, they are far from fully understood

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Regularity for list decoding Lemma: for any code, any received word can be replaced by a “low complexity” received word, which is indistinguishable from the code perspective Similar to the Frieze-Kannan weak regularity Viewpoint: codewords are “tests”

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Decompositions of polynomials The dichotomy theorem can be applied iteratively, to decompose any low-degree polynomial as a function of a few polynomials which are pseudo-random To a large extent, pseudo-random polynomials behave as “independent variables” Made precise in higher-order Fourier analysis

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Follow up work We extend the current result to the case of large fields This requires a few new ingredients: 1.Optimizing the arguments, to get a polynomial dependency on the field size 2.Extending higher-order Fourier to large fields, with bounds independent of field size

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Take home message Notions of structure and pseudo-randomness are very powerful; dichotomy theorems make them universal This work: coding theory, applied to RM codes Other applications: math - graph theory, number theory, ergodic theory, discrete geometry; CS - property testing, complexity, algorithms Question: do our techniques generalize to other codes? Other domains?